Ann. Scient. Éc. Norm. Sup.
4
e
série, t. 48, 2015, p. 409 à 451
THE GEOMETRIC SATAKE CORRESPONDENCE
FOR RAMIFIED GROUPS
X ZHU
A. – We prove the geometric Satake isomorphism for a reductive group defined over
F = k((t)), and split over a tamely ramified extension. As an application, we give a description of the
nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.
R. – Nous démontrons l’isomorphisme de Satake géométrique pour un groupe réductif dé-
fini sur F = k((t)) et déployé sur une extension modérément ramifiée. Nous donnons comme applica-
tion une description des cycles évanescents sur certaines variétés de Shimura via les modèles locaux de
Rapoport-Zink-Pappas.
Introduction
The Satake isomorphism (for unramified groups) is the starting point of the Langlands
duality. Let us first recall its statement. Let F be a non-Archimedean local field with ring
of integers O and residue field k, and let G be a connected unramified reductive group
over F (e.g., G = GL
n
). Let A ⊂ G be a maximal split torus of G, and W
0
be the
Weyl group of (G, A). Let K be a hyperspecial subgroup of G(F ) containing A( O) (e.g.,
K = GL
n
( O)). Then the classical Satake isomorphism describes the spherical Hecke algebra
Sph = C
c
(K \ G(F )/K), the algebra of compactly supported bi-K-invariant functions
on G(F ) under convolution. Namely, there is an isomorphism of algebras
Sph ' C[X
•
(A)]
W
0
,
where X
•
(A) is the coweight lattice of A, and C[X
•
(A)]
W
0
denotes the W
0
-invariants of the
group algebra of X
•
(A).
If F has positive characteristic p > 0, then the classical Satake correspondence has a vast
enhancement. For simplicity, let us assume that G is split over F (for the general case, see
Theorem A.12). Let us write G = H ⊗
k
F for some split group H over k so that K = H( O).
Let Gr
H
= H(F )/H( O) be the affine Grassmannian of H. Choose a prime different from p,
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
0012-9593/02/© 2015 Société Mathématique de France. Tous droits réservés
410 X. ZHU
and let Sat
H
be the category of (K ⊗
¯
k)-equivariant perverse sheaves with Q
`
-coefficients
on Gr
H
⊗
¯
k. Then this is a Tannakian category and there is an equivalence
Sat
H
' Rep(G
∨
Q
`
),
where G
∨
Q
`
is the dual group of G and Rep(G
∨
Q
`
) is the tensor category of algebraic represen-
tations of G
∨
Q
`
(cf. [10, 19]).
There is also a version of Satake isomorphism for an arbitrary reductive group over F ,
as recently proved by Haines and Rostami (cf. [12])
(1)
. Namely, let B(G) be the Bruhat-Tits
building of G and v ∈ B(G) be a special vertex. Let K
v
⊂ G(F ) be the special parahoric
subgroup of G(F ) corresponding to v. Let A be a maximal split F -torus of G such that
K
v
⊃ A( O), let M be the centralizer of A in G and W
0
= N
G
(A)/M be the Weyl group
as before. Let M
1
be the unique parahoric subgroup of M(F ), and Λ
M
= M(F )/M
1
, which
is a finitely generated Abelian group. Then
(0.1) C
c
(K
v
\G(F )/K
v
) ' C[Λ
M
]
W
0
.
More explicitly, suppose that G is quasi-split so that M = T is a maximal torus. Then
Λ
M
= (X
•
(T )
I
)
σ
,
where I is the inertial group and σ is the Frobenius, and (X
•
(T )
I
)
σ
denotes the σ-invariants
of the I-coinvariants of the group X
•
(T ).
The goal of this paper is to provide a geometric version of the above isomorphism when
F has positive characteristic p and the group G is quasi-split and splits over a tamely ramified
extension. More precisely, let k be an algebraically closed field and let 6= char k be a prime.
Let G be a group over the local field F = k((t)) (so that G is quasi-split automatically), which
is split over a tamely ramified extension. That is, there is a finite extension
˜
F /F such that
G
˜
F
is split and char k - [
˜
F : F ]. Let v ∈ B(G) be a special vertex in the building of G
and let G
v
be the parahoric group scheme over O = k[[t]] (in the sense of Bruhat-Tits),
determined by v. We write LG for the loop space of G and K
v
= L
+
G
v
for the jet space
of G
v
. By definition, for any k-algebra R, LG(R) = G(R
ˆ
⊗
k
F ) and K
v
(R) = G
v
(R
ˆ
⊗
k
O).
Let
F
v
= LG/K
v
be the (twisted) affine flag variety
(2)
, which is an ind-scheme over k. Let P
v
= P
K
v
( F
v
)
be the category of K
v
-equivariant perverse sheaf on F
v
, with coefficients in Q
`
. Let H be a
split Chevalley group over Z such that G ⊗
F
F
s
' H ⊗ F
s
, where F
s
is a (fixed) separable
closure of F . Then there is a natural action of I = Gal(F
s
/F ) on H
∨
:= H
∨
Q
`
(preserving a
fixed pinning).
T 0.1. – The category P
v
has a natural tensor structure. In addition, as tensor
categories, there is an equivalence
R S : Rep((H
∨
)
I
) ' P
v
,
(1)
There is another version, known earlier, as in [6].
(2)
One would call F `
v
the affine Grassmannian of G. However, we reserve the name “affine Grassmannian” of G
for another object, as defined in Definition A.2.
4
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SÉRIE – TOME 48 – 2015 – N
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THE GEOMETRIC SATAKE CORRESPONDENCE FOR RAMIFIED GROUPS 411
such that H
∗
◦ R S is isomorphic to the forgetful functor, where H
∗
is the hypercohomology
functor.
This theorem can be regarded as a categorification of (0.1) in the case when k is alge-
braically closed and the group splits over a tamely ramified extension of k((t)). For the
description of (H
∨
)
I
when H is absolutely simple and simply-connected, see § 4.
Let us point out the following remarkable facts when the group is ramified. First, the
group (H
∨
)
I
is not necessarily connected as is shown in Remark (4.4). Second, it is well-
known that if G is unramified over F , then all the hyperspecial subgroups of G are conjugate
under G
ad
(F ) ([27, §2.5]), where G
ad
is the adjoint group of G. However, this is no longer
true for special parahoric of G if G is ramified. An example is given by the odd ramified
unitary similitude group GU
2m+1
. There are essentially two types of special parahorics
of GU
2m+1
, as given in (7.1). One of them has reductive quotient GO
2m+1
(denoted by G
v
0
),
and the other has reductive quotient GSp
2m
(denoted G
v
1
). Accordingly, the geometry
of the corresponding flag varieties F
v
0
and F
v
1
are very different, while P
v
0
' P
v
1
.
Indeed, their Schubert varieties (i.e., closures of K
v
i
-orbits) are both parameterized by
irreducible representations of GO
2m+1
. Let F
v
0
¯µ
2m,1
(resp. F
v
1
¯µ
2m,1
) be the Schubert
variety in F
v
0
(resp. F
v
1
) parameterized by the standard representation of GO
2m+1
.
Then it is shown in [31] that F
v
0
¯µ
2m,1
is not Gorenstein, while in [25] that F
v
1
¯µ
2m,1
is
smooth. On the other hand, the intersection cohomology of both varieties gives the standard
representation of GO
2m+1
. In addition, the stalk cohomologies of both sheaves are the
“same”. See Theorem 0.3 below.
R 0.1 . – Instead of considering a special parahoric K
v
of LG, one can begin with
the special maximal “compact” K
0
v
, (i.e., K
0
v
= L
+
G
0
v
, where G
0
v
is the stabilizer group
scheme of v as constructed by Bruhat-Tits), and consider the category of K
0
v
-equivariant
perverse sheaves on LG/K
0
v
. However, from a geometric point of view, this is less natural
since K
0
v
is not necessarily connected and the category of K
0
v
-equivariant perverse sheaves
is complicated. In fact, we do not know how to relate this category to the Langlands dual
group yet. In addition, when we discuss the Langlands parameters in Section 6, it is also more
“correct” to consider K
v
rather than K
0
v
.
The idea of the proof of the theorem is as follows. Using Gaitsgory’s nearby cycle functor
construction as in [8, 31], we construct a functor
Z : Sat
H
→ P
v
,
which is a central functor in the sense of [2]. By standard arguments in the theory of
Tannakian equivalence and the Mirkovic-Vilonen theorem, this already implies that
P
v
' Rep(
˜
G
∨
) for certain closed subgroup
˜
G
∨
⊂ H
∨
. Then we identify
˜
G
∨
with (H
∨
)
I
using the parametrization of the K
v
-orbits on F
v
.
R 0.2 . – (i) We believe that the same argument (maybe with small modifications)
should work for groups split over wild ramified extensions. However, we have not checked
this carefully.
(ii) Our approach is more inspired by [8] rather than [19]. However, it would be interesting
to know whether there is the similar theory of MV-cycles in the ramified case. It seems that
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
412 X. ZHU
the geometry of semi-infinite orbits on F
v
is similar to the unramified case, except when
F
v
corresponds to one type of special parahorics for odd unitary groups (the one denoted
by G
v
1
as above). We do not know what happens in this last case.
(iii) Our theorem and the method also share the similar features with the results of Nadler
on geometric Satake for real groups [20].
When the group G is quasi-split over the non-Archimedean local field F = F
q
((t)) and
v is a special vertex of B(G, F ), the affine flag variety F
v
is defined over F
q
. We assume that
v is very special, i.e., it remains special when we base change G to F
q
((t)) (see § 6 for more dis-
cussions of this notion). Then we can consider the category of K
v
-equivariant semi-simple
perverse sheaves on F
v
, pure of weight zero, and denote it by P
0
v
. On the other hand, let I
be the inertial group of F and σ be the Frobenius of Gal(k/F
q
), where k = F
q
. Then the ac-
tion of Gal(F
s
/F ) on H
∨
(via the pinned automorphisms) induces a canonical action of σ
on (H
∨
)
I
, denoted by act
alg
. One can form the semidirect product (H
∨
)
I
o
act
alg
Gal(k/F
q
),
which can be regarded as a proalgebraic group over Q
`
, and consider the category of alge-
braic representations of (H
∨
)
I
o
act
alg
Gal(k/F
q
), denoted by Rep((H
∨
)
I
o
act
alg
Gal(k/F
q
)).
T 0.2. – In this case, the functor R S in Theorem 0.1 can be extended to an
equivalence
R S : Rep((H
∨
)
I
o
act
alg
Gal(k/F
q
)) ' P
0
v
,
whose composition with H
∗
is isomorphic to the forgetful functor.
Let us mention that under this equivalence, the restriction to Gal(k/F
q
) of the represen-
tation (H
∨
)
I
o
act
alg
Gal(k/F
q
) on H
∗
( F ) for F ∈ P
0
v
is NOT the natural Galois action
of Gal(k/F
q
) on H
∗
( F ). However, their difference can be described explicitly. See Section 4
and appendix for more details.
Our next result is to use the ramified geometric Satake isomorphism to obtain the stalk
cohomology of sheaves on F
v
(i.e., the corresponding Lusztig-Kato polynomial in ramified
case), following an idea of Ginzburg (cf. [10]). Let us state the result precisely. The centralizer
of A in our case is a maximal torus of G, denoted by T . Then the K-orbits on F
v
are labeled
by X
•
(T )
I
/W
0
, W
0
-orbits of the coinvariants of the cocharacter group of T . For ¯µ ∈ X
•
(T )
I
,
let
˚
F
v ¯µ
be the corresponding orbit. For a representation V of (H
∨
)
I
, let V (¯µ) be the weight
space of V for (T
∨
)
I
. Let X
∨
∈ Lie(H
∨
)
I
be a certain principal nilpotent element (see
Section 5 for the details), which induces a filtration F
i
V (¯µ) = (ker X
∨
)
i+1
∩V (¯µ) on V (¯µ),
called the Brylinski-Kostant filtration. Then we have
T 0.3. – For V ∈ Rep((H
∨
)
I
), let R S(V ) ∈ P
v
be the corresponding sheaf. Then
dim H
2i−(2ρ,¯µ)
R S(V )|
F `
v ¯µ
= dim gr
F
i
V (¯µ).
Here H
∗
denotes the cohomology sheaves, and 2ρ is the sum of positive roots of H, see
Section 1 for the meaning of (2ρ, ¯µ).
One of our main motivations of this work is to apply these results to the calculation of the
nearby cycles of certain ramified unitary Shimura varieties, via the Rapoport-Zink-Pappas
local models. For example, we obtain the following theorem (see Section 7 for details).
4
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SÉRIE – TOME 48 – 2015 – N
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THE GEOMETRIC SATAKE CORRESPONDENCE FOR RAMIFIED GROUPS 413
T 0.4. – Let G = GU(r, s) be a unitary similitude group associated to an imagi-
nary quadratic extension F/Q and a Hermitian space (W, φ) over F/Q. Let p > 2 be a prime
where F/Q is ramified and the Hermitian form is split. Let K
p
be a special parahoric subgroup
of G = G(Q
p
). Let K = K
p
K
p
⊂ G(Q
p
)G(A
p
f
) be a compact open subgroup with K
p
small
enough. Let Sh
K
be the associated Shimura variety over the reflex field E and Sh
K
p
be the in-
tegral model of Sh
K
over O
E
p
(as defined in [23]). Then for 6= p, the action of the inertial
subgroup I of Gal(Q
p
/F
p
) on the nearby cycle Ψ
Sh
K
p
⊗
O
E
p
O
F
p
(Q
`
) is trivial.
By applying Theorem 0.3, it will not be hard to determine the traces of Frobenius on these
sheaves explicitly, which will be the input of the Langlands-Kottwitz method to determine
the local Zeta factors of Sh
K
. Instead, we characterize these traces of Frobenius in terms of
Langlands parameters, which verifies a conjecture of Haines and Kottwitz in this case (see
Proposition 7.4).
R 0.3 . – (i) While the definition of the integral model of a PEL-type Shimura
variety at an “unramified” prime p (i.e., the group is unramified at p and K
p
is hyperspe-
cial) is well-known (cf. [15]), the definition of such a model at the ramified prime p (even
for K
p
special) is a subtle issue. In [21, 23], the integral models Sh
K
p
are defined as certain
closed subschemes of certain moduli problems of Abelian varieties. Except a few cases
(e.g., (r, s) = (n −1, 1) and n = r + s is small), there is no moduli description of Sh
K
p
so far. In general, Sh
K
p
are not smooth. Indeed, as shown in [21, 23], when n = r + s is
odd and (r, s) = (n −1, 1), for the special parahoric K
p
of G(Q
p
) with reductive quotient
GO
n
, Sh
K
p
is not even semi-stable.
(ii) If r 6= s, then we know that E = F and the above theorem gives a complete description
of the monodromy on the nearby cycles of Sh
K
p
. If r = s, then E = Q, and the complete
description of the monodromy is more complicated. See Section 7 for details. In any case, the
action of inertia on the nearby cycle is semi-simple.
(iii) We hope that there will be a “good” compactification of such Shimura varieties Sh
K
p
.
Then the above theorem, together with the existence of such compactification, would imply
that the monodromy of H
∗
c
(Sh
K
⊗
E
p
F
p
) is trivial.
(iv) The triviality of the monodromy as above would have the following surprising conse-
quence for the Albanese of Picard modular surfaces. Namely, in the case when (r, s) = (2, 1),
F/Q is ramified at p > 2 and K
p
= G(Q
p
) is a special parahoric, the Albanese Alb(Sh
K
p
)
of Sh
K
p
is trivial. It will be interesting to find the “optimal” level structure at p so that
Alb(Sh
K
p
) can be possibly non-trivial. More detailed discussion will appear elsewhere.
Let us quickly describe the organization of the paper. We will prove Theorem 0.1 and
Theorem 0.2 in §1-4. Then we prove Theorem 0.3 in §5.
In § 6, we briefly discuss the Langlands parameters associated to a smooth representation
of a quasi-split p-adic group, which has a vector fixed by a special parahoric. We call them
“spherical” representations, and we will see that their Langlands parameters can be described
easily. Again, the correct point of view is to consider the special parahoric rather than the
special maximal compact. Then in § 7, we apply the previous results to study the nearby cycles
on certain unitary Shimura varieties.
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE